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Epidemic Models Practical 2 (Lecture 7): Deterministic Compartmental Models in Populations with Different Levels of Susceptibility and Infectivity - 1.5 Hours

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Epidemic Models Practical 2 (Lecture 7): Deterministic Compartmental Models in Populations with Different Levels of Susceptibility and Infectivity - 1.5 Hours Epidemic Models Practical 2 (Lecture 7): Deterministic compartmental models in populations with different levels of susceptibility and infectivity - 1.5 hours Accompanying R codes: EpiTut2a.R (Case 1), EpiTut2b.R (Case 2), EpiTut2b_init.R (optional section) with corresponding ODEs coded in SIR.R, SIRMajorSus.R, SIRMajorSusInf.R and function for R0 and WAIFW-matrix in betaAndR0fc.R Notes & Tips: 1. Move all of the above R-files into your working directory and change the directory path in the R- codes for EpiTut2a.R and Epitut2b.R 2. Skip the mathematical derivations in the ‘analytical assessment’ sections if you feel uncomfortable with mathematical equations Outline: In this practical you will learn how to use deterministic epidemiological models for predicting epidemiological characteristics and the impact of control strategies in populations with variation in susceptibility. In particular, we explore how distinct levels of susceptibility affect epidemiological characteristics for different scenarios: o Case 1: only difference in susceptibility o Case 2: susceptibility confers difference in infectivity 1. Case 1: Difference in susceptibility only Figure 1 shows a flow diagram for an SIR model in a closed population of constant size N for a disease for which individuals belong to one of two categories according to their susceptibility to the infection. Let S+ and S- denote the ratio of susceptible individuals with high and low susceptibility, respectively, with corresponding transmission rates β+ = β and β- = εβ, with 0 ≤ 휀 ≤ 1. Let’s further assume that both groups have equal recovery rate γ and that the proportion of individuals with low susceptibility is p, 0 ≤ 푝 ≤ 1. As in previous applications, we assume that the contact structure is homogeneous. 1.1 Empirical assessment: a) Give some (real life) examples for which the model in Figure 1 may be a valid representation. b) What value does the parameter ε have if the presence of a gene confers complete resistance to infection? What is the value of ε in homogeneous populations? 1.2 Analytical assessment: a) Use the formula for R0 below to obtain the minimum proportion of individuals with low susceptibility required for preventing an epidemic outbreak as a function of the model parameters β,γ,ε,p. ퟏ b) Show that in the case of complete resistance this proportion is equal to ퟏ − where 푹̂ퟎ 푹̂ퟎ refers to the basic reproductive ratio in a homogeneous population with transmission rate β. c) What proportion of fully resistant individuals would be required to prevent an outbreak for a disease with an estimated value of R0=2 in a population with high susceptibility? 1 푅 = (푝휀훽 + (1 − 푝)훽) 0 훾 1.3 Numerical assessment of the infection dynamics: Open the R-script EpiTut2a.R and auxiliary file SIRMajorSus.R The code describes the above SIR model in heterogeneous populations such as those described above with parameters β = 0.8, γ = 1/4, ε=0.3 and p = 0.2 in a population of N=1000 individuals, where 1 infectious individual of the type with low susceptibility is introduced into a population of 999 non-infected individuals. Take some time to familiarize yourself with the code. Then use the code (in particular steps 1- 3) to a) calculate R0 for the above parameters. What is the required proportion of individuals with low susceptibility to prevent a disease outbreak? b) generate infection profiles for the above population and compare them with those corresponding to a homogeneous population with the same average transmission rate. Does ignoring heterogeneity cause an upward or downward bias for the total proportion of infected individuals at the end of the epidemic? c) Explore the profiles for different parameter combinations. Does heterogeneity in susceptibility always produce an upward / downward bias? 1.4 Optional section: d) Set the parameter values back to the original values (see above). Use steps 4 and 5 in the R-code to explore the impact of the effect size ε on the epidemiological characteristics. Looking at the graphs you generated, how much do peak prevalence and duration of the epidemics differ between the scenarios where the maximum transmission rate β is reduced by a factor ε = 0, ε = 0.5 and ε = 1 for a proportion p=0.2 of individuals with low susceptibility? Are 20% of completely resistant individuals sufficient to prevent an epidemic? e) Similarly, what impact does the proportion of individuals with low susceptibility have on the epidemiological characteristics in the case that the transmission rate for the highly susceptible group is twice as high as that of the group with low susceptibility and parameter values for β, γ as above? Can the presence of individuals with low susceptibility prevent an outbreak? 2. Case 2: Difference in susceptibility and infectivity In the model above we assumed that individuals differ in susceptibility only. Let’s now relax this assumption and explore how differences in both susceptibility and infectivity affect the epidemics. For simplicity we assume that differences in susceptibility may confer differences in infectivity, and that S+ and I+ denote the category of susceptible and infectious individuals from the high susceptibility group and S- and I- denote the corresponding categories for the low susceptibility group. The figure below illustrates the flow chart of the corresponding epidemiological model. As above we assume that both groups have equal recovery rate γ and that the proportion of individuals with low susceptibility is p, with 0 ≤ 푝 ≤ 1. For the analysis it is helpful to re-parameterize the above model by defining the transmission rates relative to a reference rate β0. Let us set β++ = β0, β-+ = εS * β0, β+- = εI * β0 and β-- = εI * εS * β0. Here the scaling factor εS, 0 ≤ 휀푆 ≤ 1 describes the effect of the low susceptibility type on the transmission rate and the factor εI describes the effect of the low susceptibility type on its infectivity. Thus, the WAIFW (beta) matrix for the above model is 훽 훽 훽 휀 훽 β = ( ++ +−) = ( 0 퐼 0 ) 훽−+ 훽−− 휀푆훽0 휀퐼휀푆훽0 2.1. Analytical assessment (a) What can be said about the value of 휀퐼 and about the relative values for the transmission parameters in the case that (i) difference in susceptibility confers no difference in infectivity (ii) susceptibility and infectivity are positively correlated (i.e. high susceptibility confers high infectivity) (iii) susceptibility and infectivity are negatively correlated (high susceptibility confers low infectivity) Can you think of biological explanations for each of the scenarios (i) to (iii)? Which scenario is most likely in your opinion? (b) Write down the model equations, the next generation matrix of the number of secondary cases produced through each of the four transmission routes, and the equations for the number of secondary cases produced by each of the two infectious classes (I+ and I-) to get expressions for the + - upper and lower bounds for R0, i.e. R0 and R0 . The actual value for R0 is the eigenvalue of the next generation matrix. We will calculate this value in the R-script below. 2.2. Numerical assessment Open the R-file EpiTut2b.R and auxiliary files SIRMajorSusInf.R and c The code describes the above SIR model in a population comprising 1000 individuals. In these simulations, parameter values are set to γ = ½ days, β0 = 1.5 per day, εS=0.5, and p = 0.2. For εI three values are chosen, representing different relationships between susceptibility and infectivity, i.e εI =0.2, εI = 1 and εI = 5, respectively. The epidemics starts by introducing 1 infectious individual of the low susceptible type into a population of 999 non-infected individuals. Take some time to familiarize yourself with the code. Then use the code (in particular steps 2 and 3) to answer the following questions. a. Check your model equations derived in 2.1b above by comparing your equations with the derivatives coded in the R-scripts (SIRMajorSusInf.R and betaAndR0fc.R). Also check your - expressions for the 2 by 2 next generation matrix of the number of secondary cases and R0 + and R0 derived in 2.1b (Step 2 in code – formulas coded in betaAndR0fc.R). b. For the parameter set above, calculate the basic reproductive ratios R0 for this model for the three scenarios εI =0.2, εI = 1 and εI = 5, respectively. Which scenario gives the highest value for R0? Is this expected? (Step 2 in code of EpiTut2b.R: call the function betaAndR0fc with different values of εI). - + c. How do these values compare with the values for the two thresholds R0 and R0 , and with the corresponding R0 for homogeneous populations with the same average transmission rate β = (1-p) * β0 + p* εS * β0? (Step 2 in code: Compare values of R0 and R0_homo). d. Generate infection profiles corresponding to the four scenarios (1) homogenous population with the above average transmission rate β (2) heterogeneous population with difference in susceptibility only (εI = 1), (3) heterogeneous population where low susceptibility confers low infectivity (εI = 0.2), (4) heterogeneous population where low susceptibility confers high infectivity (εI = 5) (Step 3 in code). Inspect the plots to fill in the following table for epidemiological characteristics: Characteristics Peak Time of peak Total Duration of R0 / Scenario prevalence prevalence proportion epidemics infected 1 2 3 4 Which of the four scenarios causes the most / least severe epidemics? e. Now use step 4 in the code to explore the effect of the proportion of individuals with low susceptibility on the epidemiological characteristics. In particular, use the figures and R0 values to determine o the effect (increasing or decreasing?) of increasing the proportion of individuals with low susceptibility on time and value of peak prevalence, total proportion infecteds, duration of the epidemics and on R0.
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